Question: Solve for $a$, $ \dfrac{a + 8}{12a - 8} = \dfrac{6}{9a - 6} + \dfrac{5}{12a - 8} $
Explanation: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $12a - 8$ $9a - 6$ and $12a - 8$ The common denominator is $36a - 24$ To get $36a - 24$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{a + 8}{12a - 8} \times \dfrac{3}{3} = \dfrac{3a + 24}{36a - 24} $ To get $36a - 24$ in the denominator of the second term, multiply it by $\frac{4}{4}$ $ \dfrac{6}{9a - 6} \times \dfrac{4}{4} = \dfrac{24}{36a - 24} $ To get $36a - 24$ in the denominator of the third term, multiply it by $\frac{3}{3}$ $ \dfrac{5}{12a - 8} \times \dfrac{3}{3} = \dfrac{15}{36a - 24} $ This give us: $ \dfrac{3a + 24}{36a - 24} = \dfrac{24}{36a - 24} + \dfrac{15}{36a - 24} $ If we multiply both sides of the equation by $36a - 24$ , we get: $ 3a + 24 = 24 + 15$ $ 3a + 24 = 39$ $ 3a = 15 $ $ a = 5$